Mathematical Sciences: Topological Methods in Quantum Field Theory and Methods of Quantum Field Theory in Topology

施瓦茨提出要研究量子场论中的各种问题。他将尝试利用量子场论中的拓扑不变量来构造三维接触流形中的接触流形和勒让德结的不变量。还将学习无限维李代数的半无限上同调以及辛超流形和接触超流形理论。这些问题与约束系统量子化的一般理论和超弦理论有关。现代物理学，量子力学和相对论，是二十世纪的产物。它是在上个世纪试图解决物质的微观结构并掌握远距离作用力、电磁学和热辐射等概念的过程中建立起来的。这些发展的数学基础统称为数学物理，从控制粒子动力学的薛定谔算符的详细分析，到试图将四种已知力统一为单一理论的统一场论。

Schwarz proposes to study various problems in quantum field theory. He will try to construct invariants of contact manifolds and of Legendre knots in three-dimensional contact manifolds by means of topological invariants from quantum field theory. There will also be a study of semi-infinite cohomology of infinite- dimensional Lie algebras and the theory of symplectic and contact supermanifolds. These problems are related to the general theory of quantization of constrained systems and to the superstring theory. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concepts of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments, collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single theory.